# Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## New subclasses of the class of close-to-convex functionsHTML articles powered by AMS MathViewer

by Pran Nath Chichra
Proc. Amer. Math. Soc. 62 (1977), 37-43 Request permission

## Abstract:

In this paper we introduce new subclasses of the class of close-to-convex functions. We call a regular function $f(z)$ an alpha-close-to-convex function if $(f(z)f’(z)/z) \ne 0$ for z in E and if for some nonnegative real number $\alpha$ there exists a starlike function $\phi (z) = z + \cdots$ such that $\operatorname {Re} \;\left [ {(1 - \alpha )\frac {{zf’(z)}}{{\phi (z)}} + \alpha \frac {{(zf’(z))’}}{{\phi ’(z)}}} \right ] > 0$ for z in E. We have proved that all alpha-close-to-convex functions are close-to-convex and have obtained a few coefficient inequalities for $\alpha$-close-to-convex functions and an integral formula for constructing these functions. Let ${\mathfrak {F}_\alpha }$ be the class of regular and normalised functions $f(z)$ which satisfy $\operatorname {Re} \;(f’(z) + \alpha zf''(z)) > 0$ for z in E. $f(z) \in {\mathfrak {F}_\alpha }$ gives $\operatorname {Re} f’(z) > 0$ for z in E provided $\operatorname {Re} \alpha \geqslant 0$. A sharp radius of univalence of the class of functions $f(z)$ for which $zf’(z) \in {\mathfrak {F}_\alpha }$ has also been obtained.
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